All three players will be unrestricted free agents July 1.
Hiller, 32, has played 326 games for the Ducks since signing as a free agent in 2007. He's won at least 26 games four times, and appeared in 26 Stanley Cup Playoff games as well. He was 29-13-7 with a .911 save percentage and 2.48 goals-against average in 2013-14.
Hiller started the playoffs this year on the bench, replaced by 24-year-old rookie Frederik Andersen. Hiller replaced Andersen twice during the first round and started the first two games of the second round, but was benched again in favor of Andersen in Game 3 then passed over for 20-year-old rookie John Gibson to start the final four games of the series.
Gibson and Andersen could compete for the starting job in 2014-15 unless Murray adds a veteran goaltender.
"[Hiller] gave us a great five years, but we have young guys pushing for that spot," Murray said. "The writing was on the wall."
Though forward Teemu Selanne's retirement was feted, the end of Koivu's tenure in Anaheim was more subdued. Koivu, who turns 40 in November, had 11 goals and 29 points this season. He spent the past five seasons in Anaheim after 12 with the Montreal Canadiens.
Like Selanne, Koivu is one of the most popular players in the history of their native Finland, and within the Ducks dressing room.
"It was the hardest decision of the three," Murray said. "He had a great career. We just have younger guys pushing for spots."
Winnik, 29, spent a lot of time on Koivu's wing in the past two seasons. He had six goals and 30 points in 76 games in 2013-14, but no goals and one point in nine of the 13 playoff games.
The Ducks finished second in the NHL standings to the Boston Bruins but lost to the Los Angeles Kings in the second round, losing 6-2 in Game 7 of the first "Freeway Series." Some of the younger players Murray mentioned are a big reason for hope in Anaheim.
Rickard Rakell, Emerson Etem, Nicolas Kerdiles, Stefan Noesen and William Karlsson up front, Shea Theodore on defense and Gibson in net give the Ducks the best collection of talent not established as NHL regulars of any of the elite teams from last season.
The Ducks also have the No. 10 pick in the 2014 NHL Draft received from the Ottawa Senators (along with Noesen and Jakob Silfverberg) for Bobby Ryan, along with their original selection at No. 24. There has been plenty of speculation about Murray being active in pursuing one the high-profile centers reportedly available for trade.
"We know what the rumors are, who is available," Murray said. "We have good young assets and draft picks. I'm not afraid to use them, but I won't put ourselves behind the eight ball."